Question: Divide the following complex numbers: $\dfrac{6 e^{\pi i / 4}}{3 e^{23\pi i / 12}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $6 e^{\pi i / 4}$ ) has angle $\frac{1}{4}\pi$ and radius 6. The second number ( $3 e^{23\pi i / 12}$ ) has angle $\frac{23}{12}\pi$ and radius 3. The radius of the result will be $\frac{6}{3}$ , which is 2. The difference of the angles is $\frac{1}{4}\pi - \frac{23}{12}\pi = -\frac{5}{3}\pi$ The angle $-\frac{5}{3}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{5}{3}\pi + 2 \pi = \frac{1}{3}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{1}{3}\pi$.